User:CompactStar/Super-pitch: Difference between revisions

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The super-pitch equivalent of [[just intonation]] is intervals of the form logb(x) for positive integers b and x. This includes all of just intonation, since all just intervals can be described as logarithms (e.g. [[3/2]] = log4(8)), in addition to some irrational numbers such as log2(3).

It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 23, 16 = 222, 25 = 52, 27 = 33, 36 = 62, and 81 = 322. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos.

It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 23, 16 = 222, 25 = 52, 27 = 33, 36 = 62, and 81 = 322. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos. This means that subgroups in super-pitch theory are made of non-perfect powers, like 2.3.5.6.7.10 for example.

== Super-pitch division ==

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 The super-pitch equivalent of [[just intonation]] is intervals of the form log<sub>b</sub>(x) for positive integers b and x. This includes all of just intonation, since all just intervals can be described as logarithms (e.g. [[3/2]] = log<sub>4</sub>(8)), in addition to some irrational numbers such as log<sub>2</sub>(3).
-It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 2<sup>3</sup>, 16 = 2<sup>2<sup>2</sup></sup>, 25 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 36 = 6<sup>2</sup>, and 81 = 3<sup>2<sup>2</sup></sup>. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos.
+It is possible to construct super-pitch equivalents of most concepts in [[regular temperament theory]]. There exists a super-pitch equivalent of prime factorization–every integer greater than 2 can be uniquely expressed as a power tower of numbers in the sequence OEIS [https://oeis.org/A007916 A007916] (non-perfect powers). For example, 8 = 2<sup>3</sup>, 16 = 2<sup>2<sup>2</sup></sup>, 25 = 5<sup>2</sup>, 27 = 3<sup>3</sup>, 36 = 6<sup>2</sup>, and 81 = 3<sup>2<sup>2</sup></sup>. From this, it is straightforward to define the super-pitch equivalent of [[monzo]]s, or "super-monzos" (just substitute prime factorization for this power tower representation). Super-[[vals]], super-[[mapping]]s, and even super-[[temperament]]s can be derived by using super-monzos instead of regular monzos. This means that subgroups in super-pitch theory are made of non-perfect powers, like 2.3.5.6.7.10 for example.
 == Super-pitch division ==